COMPLEMENTATION AND EMBEDDINGS OF c₀(I) IN BANACH SPACES

Abstract

We investigate in this paper the complementation of copies of $c_0(I)$ in some classes of Banach spaces (in the class of weakly compactly generated (WCG) Banach spaces, in the larger class $\mathcal{V}$ of Banach spaces which are subspaces of some $C(K)$ space with $K$ a Valdivia compact, and in the Banach spaces $C([1, \alpha ])$, where $\alpha$ is an ordinal) and the embedding of $c_0(I)$ in the elements of the class $\mathcal{C}$ of complemented subspaces of $C(K)$ spaces. Two of our results are as follows:

(i) in a Banach space $X \in \mathcal{V}$ every copy of $c_0(I)$ with $\# I < \aleph _{\omega}$ is complemented;

(ii) if $\alpha _0 = \aleph _0$, $\alpha _{n+1} = 2^{\alpha _n}$, $n \geq 0$, and $\alpha = \sup \{\alpha _n : n \geq 0\}$ there exists a WCG Banach space with an uncomplemented copy of $c_0(\alpha )$.

So, under the generalized continuum hypothesis (GCH), $\aleph _{\omega}$ is the greatest cardinal $\tau$ such that every copy of $c_0(I)$ with $\# I < \tau$ is complemented in the class $\mathcal{V}$. If $T : c_0(I) \to C([1,\alpha ])$ is an isomorphism into its image, we prove that:

(i) $c_0(I)$ is complemented, whenever $\| T \| ,\| T^{-1} \| < (3/2)^{\frac 12}$;

(ii) there is a finite partition $\{I_1, \dots , I_k\}$ of $I$ such that each copy $T(c_0(I_k))$ is complemented.

Concerning the class $\mathcal{C}$, we prove that an already known property of $C(K)$ spaces is still true for this class, namely, if $X \in \mathcal{C}$, the following are equivalent:

(i) there is a weakly compact subset $W \subset X$ with ${\rm Dens}(W) = \tau$;

(ii) $c_0(\tau )$ is isomorphically embedded into $X$.

This yields a new characterization of a class of injective Banach spaces.

2000 Mathematical Subject Classification: 46B20, 46B26.